High-order coupling high-temperature superconducting dual-passband filter

Abstract

The invention relates to the technical field of dual-passband filters, and discloses a high-order coupling high-temperature superconducting dual-passband filter and a design method thereof. According to the invention, a 12-order high-temperature superconducting dual-passband filter is constructed, the filter has three pairs of symmetrical real number transmission zero points, and equal ripple response is realized between sub-passbands and in the sub-passbands; and the device has the advantages of simple structure and small size.

Description

Technical Field

[0001] The present invention relates to the technical field of dual-passband filters, and in particular to a high-order coupled high-temperature superconducting dual-passband filter and a design method thereof. Background Art

[0002] In the construction of satellite communication and deep space exploration systems, it is necessary to cover a variety of signal formats and working modes to ensure the comprehensiveness of system functions, so as to meet the needs of various application scenarios and fields. Using a multi-channel transceiver system with multi-band working capability to replace multiple single-channel transceiver systems and combining multiple wireless communication receiving circuits into one can not only "multiply" reduce the size of the wireless communication system, but also be the main solution to solve the multi-frequency communication system's requirements for multi-band sharing, miniaturization and low power consumption of microwave hardware. Among them, multi-band microwave filters, as a key technology, have become the core component to meet these requirements. In the design of multi-band filters, dual-band filters (DBF) are a typical representative and are the most widely used. Dual-band filters can efficiently isolate and filter out interference signals in the same system by processing two independent frequency bands at the same time. The design method of dual-band filters can not only be directly applied to the development of dual-band filters themselves, but also provide important theoretical support and technical means for the design of more complex multi-band filters.

[0003] In the comprehensive design method of dual-passband filters, there are mainly two common technical paths: single-passband transformation method and optimization method. The single-passband transformation method provides a simple and effective design solution by extending the design concept and structure of single-band filters to dual-band. This method has been widely used in multiple engineering practices due to its easy implementation and clear theory. However, with the continuous improvement of technical requirements and the increase in the complexity of filter design, the inefficiency of the single-passband transformation method has gradually become apparent. Therefore, the design method based on the optimization method has gradually become the mainstream. The optimization method achieves the ideal frequency response by adjusting the design parameters of the filter, such as the position of the poles and zeros. In addition, with the continuous updating and optimization of EDA (electronic design automation) simulation software, the design space can be efficiently searched to find the optimal design solution. Therefore, the filter design method based on the optimization method has become more and more mature and convenient in engineering applications, and is more suitable for the needs of multi-parameter optimization such as multi-passband filters, which can improve the accuracy and efficiency of multi-passband filter design. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a high-order coupled high-temperature superconducting dual-passband filter and a design method thereof.

[0005] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0006] In a first aspect, the present invention provides a high-order coupled high-temperature superconducting dual-passband filter, comprising:

[0007] a first signal terminal, a first resonator, a second resonator, a third resonator, a fourth resonator, a fifth resonator, a sixth resonator, a seventh resonator, an eighth resonator, a ninth resonator, a tenth resonator, an eleventh resonator, a twelfth resonator, and a second signal terminal;

[0008] The first signal end, the first resonator, the second resonator and the fourth resonator are positively coupled and connected in sequence;

[0009] The fourth resonator, the fifth resonator, the eighth resonator and the ninth resonator form a four-corner cross-coupling structure; the fourth resonator, the fifth resonator, the eighth resonator and the ninth resonator are positively coupled and connected in sequence; the fourth resonator is negatively coupled and connected to the ninth resonator; the fourth resonator is positively coupled and connected to the third resonator; the fifth resonator is positively coupled and connected to the sixth resonator; the eighth resonator is positively coupled and connected to the seventh resonator; the ninth resonator is positively coupled and connected to the tenth resonator;

[0010] The ninth resonator, the eleventh resonator, the twelfth resonator and the second signal end are positively coupled and connected in sequence.

[0011] In one implementation, the first resonator and the second resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the first resonator and the second resonator are relatively arranged to form a relative coupling structure.

[0012] In one implementation, the third resonator and the tenth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with a concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with a concave rectangular fold line; the openings of the first resonator and the second resonator are arranged back to back to form a back-to-back coupling structure.

[0013] In one implementation, the fourth resonator and the ninth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with a concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with a concave rectangular fold line; the openings of the fourth resonator and the ninth resonator are arranged opposite to each other, and a relative coupling structure is formed through cross-coupling lines.

[0014] In one implementation, the fifth resonator and the eighth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the fifth resonator and the eighth resonator are arranged back to back to form a back-to-back coupling structure.

[0015] In one implementation, the sixth resonator and the seventh resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the sixth resonator and the seventh resonator are relatively arranged to form a relative coupling structure.

[0016] In one implementation, the eleventh resonator and the twelfth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the eleventh resonator and the twelfth resonator are relatively arranged to form a relative coupling structure.

[0017] In one implementation, the line width and opening width of the first resonator, the second resonator, the third resonator, the fourth resonator, the fifth resonator, the sixth resonator, the seventh resonator, the eighth resonator, the ninth resonator, the tenth resonator, the eleventh resonator and the twelfth resonator are the same.

[0018] In a second aspect, the present invention proposes a design method for the above-mentioned high-order coupled high-temperature superconducting dual-passband filter, comprising the following steps:

[0019] According to the design index of the filter, the direct synthesis method is used to solve the symmetrical filter polynomial.

[0020] According to the symmetrical filter polynomial and the circuit structure of the filter, the folded coupling matrix is ​​synthesized by using the transverse matrix method.

[0021] According to the center frequency of the filter, the absolute bandwidth of the frequency domain is calculated by using the frequency domain transformation method, and the coupling matrix is ​​optimized and adjusted;

[0022] Construct different coupling topologies and determine the design parameters of the circuit structure based on the coupling matrix;

[0023] The relative positions between the resonators are determined according to the design parameters of the circuit structure to obtain the final filter.

[0024] In one implementation, according to the design index of the filter, a direct synthesis method is used to solve the symmetric filter polynomial, including:

[0025] Determine the order of the filter based on the number of passbands, the band edge frequencies, the number of reflection zeros in each sub-passband and the total number of reflection zeros;

[0026] According to the number and position of transmission zeros, a numerator polynomial of the transmission function is constructed;

[0027] According to the initial frequency position of the reflection zero point in each sub-passband, a reflection function numerator polynomial is constructed;

[0028] Calculate the characteristic function according to the transmission function numerator polynomial and the reflection function numerator polynomial, and solve all the inflection point frequencies;

[0029] Calculate the characteristic function value according to the inflection point frequency and the band edge frequency of each sub-passband;

[0030] Select the frequency inflection point value for updating according to the inflection point distribution characteristics and alternation characteristics;

[0031] Reconstruct the transmission function numerator polynomial and the reflection function numerator polynomial using the updated frequency inflection point value, and re-solve all new inflection point frequencies;

[0032] Determine whether the difference between the characteristic function values ​​corresponding to all new inflection point frequencies in each independent sub-passband meets the set conditions; if so, the process ends; otherwise, recalculate the characteristic function value for iteration.

[0033] The present invention has the following beneficial effects:

[0034] The present invention constructs a 12-order high-temperature superconducting dual-passband filter, which has three pairs of symmetrical real transmission zeros, realizes equal ripple response between and within sub-passbands, and has the advantages of simple structure and miniaturized volume. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is a schematic diagram of the structure of a high-order coupled high-temperature superconducting dual-passband filter;

[0036] Figure 2 A schematic diagram of a design method for a high-order coupled high-temperature superconducting dual-passband filter;

[0037] Figure 3 It is a schematic diagram of the characteristic function of symmetrical dual passband with equal ripple coefficient;

[0038] Figure 4 Schematic diagram of the topology of an 8th-order dual-passband filter;

[0039] Figure 5 It is a schematic diagram of the topological structure of a 12th-order dual-passband filter;

[0040] Figure 6 It is a schematic diagram of the 12th-order dual-passband prototype function response;

[0041] Figure 7 Schematic diagram of the relationship between the spacing of different coupling structures and the offset distance between the opposite sides and the coupling coefficient;

[0042] Figure 8 It is a schematic diagram of the frequency response S number and group delay characteristics;

[0043] Fig. 9 This is a real photo of a high-order coupled high-temperature superconducting dual-passband filter;

[0044] Fig.10 Schematic diagram of the measured S-parameters and group delay. DETAILED DESCRIPTION

[0045] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0046] like Figure 1 The embodiment of the present invention provides a high-order coupled high-temperature superconducting dual-passband filter, comprising:

[0047] a first signal terminal, a first resonator, a second resonator, a third resonator, a fourth resonator, a fifth resonator, a sixth resonator, a seventh resonator, an eighth resonator, a ninth resonator, a tenth resonator, an eleventh resonator, a twelfth resonator, and a second signal terminal;

[0048] The first signal end, the first resonator, the second resonator and the fourth resonator are positively coupled and connected in sequence;

[0049] The fourth resonator, the fifth resonator, the eighth resonator and the ninth resonator form a four-corner cross-coupling structure; the fourth resonator, the fifth resonator, the eighth resonator and the ninth resonator are positively coupled and connected in sequence; the fourth resonator is negatively coupled and connected to the ninth resonator; the fourth resonator is positively coupled and connected to the third resonator; the fifth resonator is positively coupled and connected to the sixth resonator; the eighth resonator is positively coupled and connected to the seventh resonator; the ninth resonator is positively coupled and connected to the tenth resonator;

[0050] The ninth resonator, the eleventh resonator, the twelfth resonator and the second signal end are positively coupled and connected in sequence.

[0051] The 12th-order high-temperature superconducting dual-passband filter provided in this embodiment has three pairs of symmetrical real transmission zeros, and achieves equal ripple response between and within the sub-passbands; and has the advantages of simple structure and miniaturized volume.

[0052] In an optional embodiment of the present invention, the first resonator and the second resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the first resonator and the second resonator are relatively arranged to form a relative coupling structure.

[0053] In an optional embodiment of the present invention, the third resonator and the tenth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the first resonator and the second resonator are arranged back to back to form a back-to-back coupling structure.

[0054] In an optional embodiment of the present invention, the fourth resonator and the ninth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with a concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with a concave rectangular fold line; the openings of the fourth resonator and the ninth resonator are arranged opposite to each other, and a relative coupling structure is formed through cross-coupling lines.

[0055] In an optional embodiment of the present invention, the fifth resonator and the eighth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the fifth resonator and the eighth resonator are arranged back to back to form a back-to-back coupling structure.

[0056] In an optional embodiment of the present invention, the sixth resonator and the seventh resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the sixth resonator and the seventh resonator are relatively arranged to form a relative coupling structure.

[0057] In an optional embodiment of the present invention, the eleventh resonator and the twelfth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the eleventh resonator and the twelfth resonator are relatively arranged to form a relative coupling structure.

[0058] In an optional embodiment of the present invention, the line width and opening width of the first resonator, the second resonator, the third resonator, the fourth resonator, the fifth resonator, the sixth resonator, the seventh resonator, the eighth resonator, the ninth resonator, the tenth resonator, the eleventh resonator and the twelfth resonator are the same.

[0059] This embodiment can achieve a dual-passband response of the filter through simple coupling between resonators.

[0060] like Figure 2 As shown, an embodiment of the present invention provides a design method for a high-order coupled high-temperature superconducting dual-passband filter in the above-mentioned embodiment 1, comprising the following steps S101 to S105:

[0061] S101, according to the design index of the filter, using a direct synthesis method to solve the symmetrical filter polynomial;

[0062] S102, according to the symmetrical filter polynomial and in combination with the circuit structure of the filter, a folded coupling matrix is ​​synthesized by using a transverse matrix method;

[0063] S103, according to the center frequency of the filter, using the frequency domain transformation method to calculate the absolute bandwidth of the frequency domain, and optimizing and adjusting the coupling matrix;

[0064] S104, constructing different coupling topological structures, and determining design parameters of the circuit structure according to the coupling matrix;

[0065] S105. Determine the relative positions between the resonators according to the design parameters of the circuit structure to obtain a final filter.

[0066] In an optional embodiment of the present invention, step S101 uses a direct synthesis method to solve the symmetric filter polynomial according to the design index of the filter, including the following steps S201 to S208:

[0067] S201, determining the order of the filter according to the number of passbands, the band edge frequency, the number of reflection zero points in each sub-passband and the total number of reflection zero points;

[0068] S202, constructing a numerator polynomial of a transmission function according to the number and position of transmission zeros;

[0069] S203, constructing a reflection function numerator polynomial according to the initial frequency position of the reflection zero point in each sub-passband;

[0070] S204, calculating characteristic functions according to the transmission function numerator polynomial and the reflection function numerator polynomial, and solving all inflection point frequencies;

[0071] S205, calculating a characteristic function value according to the inflection point frequency and the band edge frequency of each sub-passband;

[0072] S206, selecting a frequency inflection point value for updating according to the inflection point distribution characteristic and the alternation characteristic;

[0073] S207, reconstructing the transmission function numerator polynomial and the reflection function numerator polynomial using the updated frequency inflection point value, and resolving all new inflection point frequencies;

[0074] S208, determining whether the difference between the characteristic function values ​​corresponding to all new inflection point frequencies in each independent sub-passband meets the set conditions; if so, the process ends; otherwise, recalculating the characteristic function value for iteration.

[0075] This embodiment adopts the synthesis method as the basis, and provides a direct synthesis technology and optimization (Direct Synthesis and Optimization Technique) method of the prototype function for the design index of multi-passband filter by analyzing the relationship characteristics of the equal ripple transfer function response and the number of passbands, the number of zero poles, etc. First, a multi-passband prototype function containing zero-frequency transmission zeros is constructed to effectively describe the basic characteristics of the dual-passband filter, including the frequency response of its passband and stopband. Then, the constraint function of the characteristic function is constructed, and the arithmetic solution of the rational transfer function of multiple parameter values ​​under multiple passband equal ripple conditions is obtained based on the characteristic function values ​​at the inflection point and the sideband frequency in the passband. On this basis, the method of iterative optimization design of dual-passband filtering is further introduced to realize the design of dual-passband filter with non-zero transmission zeros. With the help of EDA simulation software, the frequency response of the filter is finely adjusted by adjusting the design parameters of the equal bandwidth filter (such as poles, zero position, bandwidth, etc.), and the optimal solution is iteratively searched and found in the complex design space to achieve the performance indicators of ripples such as reflection in the passband and transfer function between sub-passbands.

[0076] Especially when combined with high-temperature superconducting technology and microstrip theory, the dual-passband filter can show the advantages of good sideband suppression, high-order coupling and equal bandwidth, and can show excellent characteristics in low-loss and ultra-narrowband signal processing. Therefore, this embodiment generates a multi-band transfer function by synthesizing an equivalent lumped element network or a coupling matrix, and designs a dual-passband high-temperature superconducting filter with equal frequency bandwidth and non-zero frequency transmission zero point. The comparison of simulation results with test results shows that the optimized high-temperature superconducting dual-passband filter not only has lower insertion loss and better out-of-band suppression characteristics, but also has equal frequency bandwidth and ripple characteristics.

[0077] In this embodiment, the dual-band filter synthesis is ultimately to obtain a feasible minimum-order rational transfer function to meet the filter's return loss, insertion loss, phase linearity and other indicators, and to produce the desired amplitude response in two passbands and three stopbands, and to produce a corresponding phase response in the passband.

[0078] For any two-port lossless filter network consisting of N coupled resonators, its normalized multi-band transmission and reflection functions can be expressed as the ratio of N-order complex polynomials. According to the single-channel filter function prototype, the transmission function S of the dual-passband filter with a given transmission zero is 21 (s) and reflection function S 11 (s), and the relationship between the polynomials E(s), F(s) and P(s) is given as follows:

[0079]

[0080] Among them, s refers to the complex frequency, expressed as s = σ + jω. ε and ε R are real constants related to the equal-ripple coefficients of the passband and stopband. F(s), P(s), and E(s) are polynomials with the leading term being one. The number and positions of the transmission zeros form the polynomial P(s), and the order of P(s) is M < N, and its roots are symmetric about the imaginary axis. The order of F(s) is N, which is determined by the initial frequency positions of the reflection zeros within each sub-passband. E(s) is a strict Hurwitz polynomial that satisfies the following equation.

[0081]

[0082] Therefore, the problem of determining the multi-passband frequency response function is transformed into the problem of determining the positions of zeros and poles. According to the above formula, the transfer function and reflection function can also be expressed as the following formula. After the characteristic function is determined, the frequency response function is determined.

[0083]

[0084] Among them, C(ω) is the redefined characteristic function, and the specific definition of this characteristic function is shown in the following formula.

[0085]

[0086] Among them, f i (i = 1,..., N) are poles and must be real numbers; z m (m = 1,..., M) are transmission zeros and can be real numbers, imaginary numbers, or complex numbers (for the case of imaginary or complex numbers, z m must be given in conjugate pairs). In the 8th-order prototype frequency-domain response of Figure 1 , the 8 in-band poles are represented by f i (i = 1,..., 8). It should be noted that f i and z m are now defined in the ω-domain. Usually, real or complex transmission zeros (in the s-domain) are predetermined and set during the approximation process.

[0087] Therefore, obtaining the transfer function polynomials E(s), F(s), and P(s) is equivalent to the synthesis process of the characteristic function. That is, by specifying the frequency ranges of the passband and stopband of the double-passband filter, the number of poles in each passband, the number of imaginary zeros M in each stopband (including the specified real or complex zeros), and the minimum return loss in each passband, the frequency-domain positions of the function poles and imaginary zeros in the best case of the double-passband filter function can be found.

[0088] Such as Figure 3The normalized prototype filter shown contains the key parameters of the filter. Given the number of sub-passbands and the band edge frequency, it can be used to determine the characteristic function C(ω). Taking a typical normalized eighth-order dual-passband filter as an example, it consists of 6 transmission zeros z m (m=1,...,6), 8 in-band reflection zeros, among which there is a transmission zero z in the first stopband and the third stopband respectively. m , there are 4 transmission zeros in the second stopband, which are used to define the suppression edge of the stopband. The lower edge frequency of the low-frequency sub-passband is set to s=-j, and the frequency range is -1~X1; the upper edge frequency of the high-frequency sub-passband is set to s=+j, and the frequency range is X2~+1. Given the number of reflection zeros N / 2 in each sub-passband, the total number of reflection zeros in all sub-passbands is the order of the filter N. According to the characteristic function of the filter, the characteristic function C(ω)=F(ω) / P(ω) is differentiated and the root is obtained. Then, by solving and analyzing the characteristics of the characteristic function and finding all the inflection point values, the optimal equal ripple response for each passband and stopband can be obtained.

[0089] Then, the root of the derivative of C(ω) is solved by the following formula to determine the critical frequency point, that is, the frequency point where C(ω) reaches an extreme value.

[0090]

[0091] exist Figure 3 A typical curve of C(ω) for an 8th order dual-band filter is given in . The figure shows two passbands and three stopbands (this passband characteristic can also be extended to multi-band filters). Let the roots of the derivative of C(ω) in a passband be c1,…,c K-1 ( Figure 3 where K = 8), where K is the number of poles in the passband. K It means that the two band edge equal ripple points given at the beginning are located near z1 and z6 respectively. Assume:

[0092] c0 <c1<···<c i < <c K-1 <c K

[0093] Each pole in the passband should be located between two consecutive c, such as Figure 3 As shown, that is, c i-1 <f i <c i First, assume the value of a pole f0, and then solve the inflection points of C(ω) on both sides (the two inflection points are quite different in value at this time |c i-1 |≠|c i|). In order to make the values ​​of the two inflection points equal, the root of the derivative of C(ω) is iteratively solved on this basis to obtain a new extreme point. At this time, the updated characteristic function can be closer to the equal ripple performance. Let C0(ω) be the initial characteristic function, and obtain:

[0094]

[0095] Assume that the updated characteristic function has a new extreme value f i ′, can replace the old value f in C0(ω) i . Assume that after multiple iterations, the obtained c i ′ is used to make C1(ω) at c i-1 and c i have the same absolute value, that is

[0096] C1(c x-1 )=-C1(c x )

[0097] This gives us the following equation:

[0098]

[0099] After further operation on the above formula, we can find f x The values ​​are:

[0100]

[0101] After all updated poles and zeros are found for each passband and stopband, an iteration is completed to obtain the updated characteristic function. The root of the derivative of C(ω) is calculated using the updated characteristic function to check the value of C(ω). The difference between the C(ω) values ​​corresponding to all new inflection frequencies will become smaller and smaller. Repeat the above iterative steps until the inflection frequency c in each sub-passband is obtained. i With the band edge frequency f i The difference between the calculated C(ω)s is approximately zero within a given accuracy range.

[0102] Similarly, the iterative solution process is also applied to the solution of the transmission zeros in the three stopbands, so that each stopband has equal ripple performance. Through the above process, the equal ripple response of the dual passband can be obtained, and this method can be applied to the solution of multiple passbands.

[0103] In this embodiment, the all-pole single-bandpass filter is completely defined by the following four parameters: order, two bandpass edge frequencies c i and ripple level RL. Similarly, in the design method discussed above, the dual-passband filter needs to consider the order N, the number of sub-passbands N i , the passband edge frequency point ci and return loss RL. However, these eight parameters are not defined independently. Figure 3 As shown, the lower edge frequency of the lowest sub-passband and the upper edge frequency of the highest sub-passband are fixed to ω = -1 and ω = +1 respectively, and the normalized transfer function of the dual-passband filter can be expressed as:

[0104]

[0105] Among them, ε1ε R1 The return loss coefficient of the first sub-passband is defined as:

[0106]

[0107] Where RL1 is the in-band echo level of the first sub-passband, expressed in dB. In an all-pole filter, function C N (ω) is represented by an N-order polynomial, where N is the sum of the orders of the two bandpass filters, i.e., N = N1 + N2. In general, an N-order polynomial is uniquely determined by N+1 points. For example Figure 1 In the dual passband filter, the polynomial C N (ω), N1 = 4, N2 = 4. When the ripple coefficient of sub-passband 1 is normalized, the ripple coefficient of sub-passband 2 is equal to ε2ε R2 / ε1ε R1 , that is:

[0108]

[0109] RL2 is the return loss level of the second band, expressed in dB. The edge frequencies are represented by c0 and c4 in the first band and by c5 and c9 in the second band. It can be observed that the frequency response of the dual passband filter is specified by eight parameters: order, number of sub-passbands, edge frequencies of the passband, and return loss, namely N1, N2, c0, c4, c5, c9, ε2ε R2 and ε1ε R1 is fixed, then the problem is equivalent to constructing an N-order polynomial from N+2 points. Figure 3In the case shown, the filter curve for N=8 has 10 corresponding points to constrain. This problem is obviously beyond the scope of the solution, and a variable reference quantity must be released to solve this problem. Therefore, for dual-passband or multi-passband filters, this embodiment can only preset the return loss value of a high-frequency sub-passband in advance, and the return loss of other passbands depends on the number and position of transmission zeros, the number of reflection zeros of each sub-passband, and the preset sub-passband bandwidth. This means that this multi-passband filter has an asymmetric characteristic (having different sub-passband bandwidths or asymmetric suppression parameters, etc.). Although the return loss within the passband of each sub-passband is equiripple, the amplitudes in different passbands are not equal or different from the values ​​set in the highest sub-passband. Therefore, there are the following two situations in which the variable parameters can be released, so that an effective solution to the transmission function can be obtained.

[0110] (1) This method can achieve an equal ripple response for the characteristic function of a symmetrical dual-passband. Figure 1 Take the frequency response in as an example. When the two sub-passbands of passband 1 and passband 2 are exactly the same, N / 2 transmission poles form one passband, and the other N / 2 transmission poles form another passband. At this time, the 8 parameters that determine the response function are reduced by half, that is, N1 = N2, c0 = c9, c4 = c5, ε2ε R2 =ε1ε R1 , so the transfer function is effectively solved. This is very valuable for high-temperature superconducting dual-passband filters, because the dual-passband systems of deep space exploration and satellite communication systems usually have the same bandwidth.

[0111] (2) If the passband ripples of different passbands are to be preset, at least one more parameter needs to be added as the filter optimization variable, allowing it to change with the optimization calculation process with the preset equal ripple return loss RL as the target. The method used is to place a transmission zero point z between the two sub-passbands. i By optimizing its position, the equal ripple return loss of the high sub-passband is consistent with the preset value of the low sub-passband. Figure 3 As shown, a transmission zero z is introduced between the two passbands. i , optimize the position of the transmission zero point f Tz After that, the required equal ripple return loss can be achieved in the sub-passband. And, if transmission zeros TZs are added between the sub-passbands, lobes between the sub-passbands can be generated. Such multiple zero positions can be used to optimize the equal ripple return loss RL and the given lobe suppression. Figure 3The 8th-order dual-passband filter shown contains four transmission zeros between the two passbands, with inflection points p1 to p3, which produce three lobes and optimize the sideband suppression level ORL in the two passbands. Similarly, each addition of a transmission zero outside the two sub-passbands will produce a corresponding lobe; these zero positions have the same effect as the transmission zero positions between the sub-passbands, and can be used for equal ripple return loss RL and the optimized ORL for a given lobe suppression.

[0112] Therefore, different equal ripple return loss and asymmetric lobe suppression in each sub-passband are pre-designed, which needs to meet the following requirements: at least one transmission zero must be used as the optimization variable between two sub-passbands of the multi-passband filter. Moreover, this method can be extended to filter design with arbitrary passband and arbitrary number of lobes.

[0113] In the above-mentioned comprehensive method, the introduction of transmission zeros as variables increases the degree of freedom of the target parameters consisting of the sub-passband bandwidth, return loss, and the number of reflection zeros and lobe suppression of each sub-passband. For a symmetrical dual-passband filter, when the number of transmission zeros M is introduced in pairs, it has the characteristics of simplicity in terms of topological structure and physical implementation structure. This embodiment proposes Figure 4 The symmetrical CQ topology shown in the figure is based on the CQ unit. The resonator R is added symmetrically at both ends of the CQ unit. ps (Resonators 4 and 5) can form transmission zeros in two sub-bands. Therefore, this structure has the characteristics of bilateral symmetry, which can reduce the workload of filter design by half. Figure 4 On the other side of the CQ unit, a pair of resonators R is introduced. ps (Resonators 4 and 5), transmission zeros can be formed in the two sub-passbands. Based on this structure, a topology of a 12th-order dual-passband filter is constructed, such as Figure 5 shown.

[0114] On the one hand, in actual engineering design, the resonant frequency f i , the coupling between the resonators M ij and the external coupling Q i , it is difficult to maintain self-consistency, and there are influences on each other. Therefore, for a fully symmetrical dual-passband filter with equal relative bandwidth, in most cases, the debugging and optimization of the target parameters may be unsolvable. On the other hand, by adjusting the position of the zero-frequency transmission zero point, adjusting the transmission zero point f Tz The position of f i =0j, the transmission zeros between the sub-passbands are dispersed in the stopband, which can achieve equal ripples in the stopband and improve the suppression of the stopband. i=0j, a symmetrical non-zero frequency transmission zero point frequency domain response is obtained, and then an iterative method is used to achieve the equal ripple response of the transfer function between the sub-passbands, such as Figure 6 shown.

[0115] According to the above analysis, the Figure 6 The filter design is carried out by using the equal ripple frequency response between the sub-passbands shown in the figure. Once the polynomials E(s), F(s) and P(s) are determined, as well as the constants ε and ε R , the coupling matrix synthesis can be performed in the same way as the conventional single-passband filter of the same order, and the multi-passband filter can be accurately reconstructed. Taking the dual-passband filter index as an example, the ranges of the low sub-passband and high sub-passband frequencies ω are set to -1 to -0.315 and 0.315 to +1 respectively, the target suppression of the three stopbands are 50dB, 40dB and 50dB respectively, and the target return loss value of the low sub-passband is 20dB, the same as that of the high sub-passband, with 3 pairs of transmission zeros. According to the initial values ​​of the six transmission zeros, the transmission function numerator polynomial P(s) can be derived, and the reflection function numerator polynomial of the dual-passband filter can also be derived by using the above method. Since the return loss of the high sub-passband is preset to 20dB, the above-mentioned alternating pole method can be used to construct the common denominator polynomial E(s). Using the inflection point frequency of out-of-band suppression, the lobe suppression can be calculated, and the evaluation function is generated by calculating the return loss at any edge frequency of the low sub-passband and the target equiripple return loss in the band. Then adjust the positions of these transmission zeros and repeat the entire optimization process until the evaluation function is infinitely close to zero. The final results of these polynomials are given in Table 1. The corresponding frequency response is shown in Figure 6 As shown, there is a pair of transmission zeros at zero frequency f i =0j, the frequency response can be almost symmetrical about the ω=0 axis, and the reflections in the two sub-passbands are equal to the ripples.

[0116] Table 1

[0117]

[0118] The N+2 folded coupling matrix can be synthesized using the lateral matrix method, and then, according to Figure 5 The CQ unit parallel resonator topology shown in the figure reconstructs the matrix. Among them, the first group of four corner elements (4-5-8-9) realizes a set of transmission zeros outside the passband, while the parallel resonant units (6 and 7) realize a non-zero frequency transmission zero pair between the sub-passbands, and the parallel resonant unit pair (3-10) at the other end of the CQ unit forms a zero-frequency transmission zero pair. This combination will minimize the coupling value of the diagonal cross-coupling in the four corner elements. In the end, all coupling values ​​and frequency offsets fully meet the actual requirements. The N+2 normalized coupling matrix obtained by the transverse matrix method is shown as follows:

[0119]

[0120] The coupling matrix has the characteristics of symmetry of the main diagonal and the secondary diagonal, the self-resonator frequency is zero, and the value of the coupling coefficient is only 8 (7 pairs + one cross coupling). The reduction of the coupling coefficient is of great significance for the simplification of filter design, debugging and optimization.

[0121] According to the above analysis, in the microstrip structure filter, the resonant frequency M of the resonator is ii , coupling coefficient M ij They affect each other, so it is difficult to achieve zero self-resonance frequency (M ii =0). Furthermore, if the equal ripple of the transfer function can be achieved in the stopband between the sub-passbands, it will have a good effect on sideband suppression. Figure 5 The self-resonant frequency M of resonators 3 and 10 in the topology ii , the separation of zero-frequency transmission zeros can be achieved. Then, the resonant frequencies of other resonators are iteratively optimized, and the resonant frequencies of the resonators are added as adjustable variables to increase a control dimension, so that equal ripple responses can be achieved between sub-passbands, and the sideband suppression between sub-passbands can be improved. The final results of the polynomials of these functions are given in Table 3, and the corresponding frequency responses are shown in Figure 6 shown.

[0122] Table 3

[0123]

[0124] The coupling matrix is ​​shown below:

[0125]

[0126] The coupling matrix is ​​symmetric along the main diagonal, and the elements M on the main diagonal are ii is no longer zero. The self-resonant frequency is not zero, that is, the 12 resonators have non-uniform resonant frequencies. For the design of microstrip structure filters, the change of the self-resonant frequency can be easily achieved by adjusting the equivalent length of the resonator, and the symmetry of the main diagonal line reduces the amount of calculation of the coupling coefficient. In this type of structure, if the resonant frequencies of all resonators are offset differently, this can not only make the reflection function within the passband and the transmission function between the sub-passbands have equal ripple reflection responses, but also maintain consistency with the topology of the initial symmetrical filter.

[0127] The coupling matrix of the dual passband filter is denormalized to obtain the general parameters required to realize the physical structure size of the path filter.

[0128] The low-pass prototype is mapped to a band-pass filter via the transformation given by:

[0129]

[0130] Among them, ω' is the normalized angular frequency in the low pass, ω is the frequency variable in the band pass, ω0 is the center frequency of the passband, △ω=ω2-ω1, ω2 and ω1 are the high-frequency sideband frequency at the high-frequency sub-passband and the low-frequency sideband frequency at the low-frequency sub-passband in the dual passband, respectively. Coupling coefficient m under physical structure ij (i≠j,I,j≠S,L) and the normalized coupling term M ij The relationship is as follows:

[0131]

[0132] Where BW represents the relative bandwidth of the bandpass filter, and f0 represents the center frequency of the bandpass filter. The quality factor of the input and output ports S and L corresponds to the normalized coupling coefficient term:

[0133]

[0134] Therefore, according to the normalized coupling coefficient obtained. Through the inverse normalization transformation, the physical coupling coefficient of the 12-order line dual-passband filter is calculated as shown in Table 4 below, and m ii ≠0(i=1,2…12)(non-same frequency resonance), m ii Less than zero means the self-resonant frequency is less than the center frequency, otherwise, the self-resonant frequency is higher than the center frequency. 49 A negative number indicates that the electromagnetic coupling characteristics are opposite to those of other couplings.

[0135] Table 4

[0136]

[0137]

[0138] In this embodiment, a resonator based on a folded open-ring half-wavelength microstrip line is designed on a YBCO / MgO / YBCO high-temperature superconducting film substrate with a thickness and a dielectric constant of 0.5 mm and 9.8, respectively. Figure 1As shown, the line width of the resonator and the width of the gap are the same, both 0.15 mm. The resonator has a meandering structure, which facilitates the coupling of adjacent resonators. This is a more practical resonator structure. When the size of the resonator is 3.3mm×3.3mm, the corresponding resonant frequency f0=3625MHz. Since the resonant frequency of the microstrip structure resonance depends on the length of the microstrip line, the resonant frequency f0 of the resonance can be controlled by adjusting the length of the resonator. By adjusting the length of the fold line extending inward from the resonator ring, the length of the half-wavelength microstrip line resonator can be adjusted.

[0139] Under the condition of extremely weak external coupling, according to the amplitude and phase characteristics of the coupling between the two resonators, the four resonators of the CQ unit meet the electromagnetic coupling conditions and phase conditions, where the cross-coupling between the CQ units is achieved using cross-coupling lines. The coupling coefficient between the two resonators can be calculated by the following formula:

[0140]

[0141] where f p1 and f p2 represents the resonant frequency of the two resonators in the electromagnetic simulation. Then, Figure 1 When the distance d between the two resonators increases, the coupling coefficient between the two resonators decreases. Conversely, the coupling coefficient increases. The relationship curves between the spacing and coupling coefficient of the three coupling structures are shown in Figure 7 As shown, the coupling coefficient between the two resonators can be adjusted by adjusting the distance between the resonators. In addition, the coupling strength can also be adjusted by adjusting the distance between the two resonators facing each other. And, the coupling strength of the resonators can be changed by adjusting the facing area between the two resonators. When the area facing each other becomes larger, the coupling strength increases, and vice versa, the coupling strength decreases. The spacing d of the three coupling structures i =0.2mm (i=1,2,3), the relationship curve between the offset distance T of the opposite sides and the coupling coefficient is as follows Figure 7 shown.

[0142] The position of the feeding port directly affects the no-load quality factor Q of the resonator. e The value can be calculated using the following formula:

[0143]

[0144] Among them, the input and output microstrip lines adopt a 50-ohm structure with a line width of 0.46 mm.

[0145] By successively calculating and extracting the spacing corresponding to the coupling coefficient between the two resonators, the location where the tap is connected to the resonator, and the coupling length of the cross-coupling line, the physical dimensions of each part of the filter can be constructed. The final layout of the entire planar circuit is as follows Figure 1 As shown in the figure, the dotted lines mark the components of the CQ unit, and Port1 and Port2 represent the signal input and output terminals, which can be connected to the SMA connector. Through simple coupling between the resonators, the dual-passband response of the filter can be achieved. The corresponding frequency response S number and group delay characteristics are shown in Figure 8 The S parameter response of the filter is in good agreement with the theoretical calculation, and the group delay can reach the expected value of ≤5ns.

[0146] Finally, the filter was fabricated on a double-sided YBCO / MgO / YBCO HTS substrate with a size of 28.73 mm × 18.99 mm (0.87 λ go ×0.58λ go ,λ go The working equivalent wavelength of the substrate with a center frequency of 3.625 GHz) is etched on one side of the HTS film by standard procedures of photolithography and ion etching, and the other side is used for grounding. The filter is then packaged in a metal shielding box. The photo of the fabricated high-order coupled high-temperature superconducting dual-passband filter is shown in Fig. 9 shown.

[0147] Fig.10 The S parameters measured at 77K are shown. The measured results show that the proposed HTS dual-passband filter has 6 transmission zeros at 3.3861GHz, 3.5906, 3.6094, 3.6454, 3.6603 and 3.8702GHz, which can well improve the sideband roll-off. The insertion loss in the two sub-passbands is 0.15dB and 0.17dB respectively, and the return loss in the passband frequency range is better than 17.8dB and 16dB respectively. The center frequency is 3625MHz, the bandwidth of the two sub-passbands is 120MHz, the band edge roll-off rate is greater than 1dB / MHz, and the suppression between the two passbands is greater than 40dB. The expected indicators can be well achieved, and the theoretical effectiveness of the direct synthesis technology and its optimization method proposed in this paper to design dual-passband filters can be verified.

[0148] The present invention provides a direct synthesis technology and an optimization method thereof to realize a comprehensive method of dual-passband / multi-passband, which is highly convenient in realizing the design and implementation of multi-passband filters with equal ripple response, equal absolute bandwidth or equal relative bandwidth. On this basis, the present invention proposes a symmetrical topological structure designed for dual-passband filters, which has the characteristics of simplicity and is very suitable for the design of microstrip filters. On the MGO substrate of double-sided YBa2Cu3O7 (YBCO) film, a typical 12th-order high-temperature superconducting dual-passband filter with three pairs of transmission zeros is designed. Finally, the results of the parameters such as the in-band insertion loss, band edge roll-off rate and group delay fluctuation within the bandwidth of the test circuit are in good agreement with the theoretical simulation values.

[0149] The present invention uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

[0150] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A high-order coupled high-temperature superconducting dual-passband filter, characterized in that: include: a first signal terminal, a first resonator, a second resonator, a third resonator, a fourth resonator, a fifth resonator, a sixth resonator, a seventh resonator, an eighth resonator, a ninth resonator, a tenth resonator, an eleventh resonator, a twelfth resonator, and a second signal terminal; The first signal end, the first resonator, the second resonator and the fourth resonator are positively coupled and connected in sequence; The fourth resonator, the fifth resonator, the eighth resonator and the ninth resonator form a four-corner cross-coupling structure; The fourth resonator, the fifth resonator, the eighth resonator and the ninth resonator are connected in a positive coupling manner in sequence; The fourth resonator is negatively coupled to the ninth resonator; The fourth resonator is positively coupled to the third resonator; The fifth resonator is positively coupled to the sixth resonator; The eighth resonator is positively coupled to the seventh resonator; The ninth resonator is positively coupled to the tenth resonator; The ninth resonator, the eleventh resonator, the twelfth resonator and the second signal end are positively coupled and connected in sequence.

2. A high-order coupled high-temperature superconducting dual-passband filter according to claim 1, characterized in that: The first resonator and the second resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the first resonator and the second resonator are arranged relative to each other to form a relative coupling structure.

3. The high-order coupled high-temperature superconducting dual-passband filter according to claim 1, characterized in that: The third resonator and the tenth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the first resonator and the second resonator are arranged back to back to form a back-to-back coupling structure.

4. The high-order coupled high-temperature superconducting dual-passband filter according to claim 1, characterized in that: The fourth resonator and the ninth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the fourth resonator and the ninth resonator are arranged opposite to each other, and a relative coupling structure is formed through cross-coupling lines.

5. The high-order coupled high-temperature superconducting dual-passband filter according to claim 1, characterized in that: The fifth resonator and the eighth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the fifth resonator and the eighth resonator are arranged back to back to form a back-to-back coupling structure.

6. The high-order coupled high-temperature superconducting dual-passband filter according to claim 1, characterized in that: The sixth resonator and the seventh resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the sixth resonator and the seventh resonator are arranged relative to each other to form a relative coupling structure.

7. The high-order coupled high-temperature superconducting dual-passband filter according to claim 1, characterized in that: The eleventh resonator and the twelfth resonator are both zigzag open-loop half-wavelength microstrip lines, the first side of the zigzag open-loop half-wavelength microstrip line is provided with an opening, the second side and the fourth side of the zigzag open-loop half-wavelength microstrip line are provided with an inward-concave T-shaped fold line, and the third side of the zigzag open-loop half-wavelength microstrip line is provided with an inward-concave rectangular fold line; the openings of the eleventh resonator and the twelfth resonator are arranged relative to each other to form a relative coupling structure.

8. The high-order coupled high-temperature superconducting dual-passband filter according to claim 1, characterized in that: The line width and opening width of the first resonator, the second resonator, the third resonator, the fourth resonator, the fifth resonator, the sixth resonator, the seventh resonator, the eighth resonator, the ninth resonator, the tenth resonator, the eleventh resonator and the twelfth resonator are the same.

9. A design method for the high-order coupled high-temperature superconducting dual-passband filter according to any one of claims 1 to 8, characterized in that: The following steps are involved: According to the design index of the filter, the direct synthesis method is used to solve the symmetrical filter polynomial. According to the symmetrical filter polynomial and the circuit structure of the filter, the folded coupling matrix is ​​synthesized by using the transverse matrix method. According to the center frequency of the filter, the absolute bandwidth of the frequency domain is calculated by using the frequency domain transformation method, and the coupling matrix is ​​optimized and adjusted; Construct different coupling topologies and determine the design parameters of the circuit structure based on the coupling matrix; The relative positions between the resonators are determined according to the design parameters of the circuit structure to obtain the final filter.

10. The design method according to claim 9, characterized in that: According to the design index of the filter, the direct synthesis method is used to solve the symmetrical filter polynomial, including: Determine the order of the filter based on the number of passbands, the band edge frequencies, the number of reflection zeros in each sub-passband and the total number of reflection zeros; According to the number and position of transmission zeros, a numerator polynomial of the transmission function is constructed; According to the initial frequency position of the reflection zero point in each sub-passband, a reflection function numerator polynomial is constructed; Calculate the characteristic function according to the transmission function numerator polynomial and the reflection function numerator polynomial, and solve all the inflection point frequencies; Calculate the characteristic function value according to the inflection point frequency and the band edge frequency of each sub-passband; Select the frequency inflection point value for updating according to the inflection point distribution characteristics and alternation characteristics; Reconstruct the transmission function numerator polynomial and the reflection function numerator polynomial using the updated frequency inflection point value, and re-solve all new inflection point frequencies; Determine whether the difference between the characteristic function values ​​corresponding to all new inflection point frequencies in each independent sub-passband meets the set conditions; if so, the process ends; otherwise, recalculate the characteristic function value for iteration.

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